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MAA Invited Paper Session: Algebraic Structures Motivated by Knot Theory Abstracts

Friday, August 7, 9:00 AM - 11:20 AM and 1:00 PM - 5:00 PM, Marriott Wardman Park, Delaware A

The area of knot theory has been developing rapidly in recent years. Most recent advances rely on the connections between algebra, homological algebra and knot theory. Examples include the Jones polynomial, topological quantum field theories, skein modules of links in 3-manifolds, Khovanov link and Heegard-Floer homologies, homology of distributive structures (i.e. quandles, racks, distributive lattices) and Yang-Baxter operators, as well as categorifications of knot polynomials and other appropriate combinatorial structures. These new developments relate knot theory to other branches of mathematics including number theory, Lie theory, statistical physics, etc, and employ tools far beyond the traditional ones from algebraic topology. These ideas mark the beginning of a new era in knot theory that includes relationships with four-dimensional problems and the creation of new forms of algebraic topology relevant to knot theory. Moreover, knot theory has numerous results and open problems requiring only knowledge of linear algebra, and are therefore accessible to undergraduates. We propose to bring together students and faculty active in these areas to share them with the broader mathematical community and encourage future collaboration and investigation.

Alissa Crans, Loyola Marymount University
Jozef Przytycki, George Washington University
Radmila Sazdanovic, North Carolina State University

Knots and Knot Theory

9:00 AM - 9:40 AM
Lou Kauffman, University of Illinois at Chicago

Knotting and weaving has been part of all cultures for thousands of years, but this subject was not studied mathematically until the middle of the 19-th century when the scientist Lord Kelvin (Sir William Thompson) conceived the Theory that atoms were knotted vortices in the "luminferous aether" (a hypothetical fluid that filled empty space). Kelvin convinced the mathematician Peter Guthrie Tait and his team to make a table of knots. At the same time other mathematicians were preparing the ground for actually doing mathematics with knots. By the turn of the century, the aether theory had disappeared, but the mathematical theory of knots was beginning to thrive! Knot theory is today an active part of mathematics, with many applications. We demonstrate problems and questions about knots by using rope, magic tricks and computer graphics. We will discuss tricks that illuminate the topology. This will include knots that are unknotted in surprising ways, the Dirac string trick that describes the quantum state of an electron, how to weave a braided belt and what this has to do with elementary particles, how the question of knotted vortices was resolved by the use of three dimensional printing. In the course of this, we shall discuss how knots are studied by using diagrams and how this way of thinking leads to new structures such as the Jones polynomial, Khovanov homology and virtual knot theory. The talk will be elementary and self-contained.

Knot Coloring: A Diagrammatic Approach to Algebraic Invariants

9:50 AM - 10:30 AM
Heather Russell, Washington College

A knot is a circle properly embedded in three-dimensional space. A central issue in knot theory is determining when two knots are the same where same means ambiently isotopic. There are many ways to algebraically tackle this question, and when we are lucky, there is a convenient diagrammatic framework encoding the algebra. As an example, we will explore how the combinatorial rules of knot coloring encode dihedral representations of the fundamental group of the knot complement. We will discuss Fox coloring as well as the much less extensively explored notion of Dehn coloring with a focus on advantages of Dehn coloring.

Topological Symmetries of Molecules

10:40 AM - 11:20 AM
Erica Flapan, Pomona College

Chemists have defined the $${\it point group}$$ of a molecule as the group of rigid symmetries of its molecular graph in $$\mathbb{R}^3$$. While this group is useful for analyzing the symmetries of rigid molecules, it does not include all of the symmetries of molecules which are flexible or can rotate around one or more bonds. To study the symmetries of such molecules, we define the $$topological symmetry group$$ of a graph embedded in $$\mathbb{R}^{3}$$ to be the subgroup of the automorphism group of the abstract graph that is induced by homeomorphisms of $$\mathbb{R}^{3}$$. This group gives us a way to understand not only the symmetries of non-rigid molecular graphs, but the symmetries of any graph embedded in $$\mathbb{R}^3$$. The study of such symmetries is a natural extension of the study of symmetries of knots. In this talk we will present a survey of results about the topological symmetry group and how it can play a role in analyzing the symmetries of non-rigid molecules.

An Introduction to Quandles

1:00 PM - 1:40 PM
Alissa Crans, Loyola Marymount University

A quandle is a set equipped with two binary operations satisfying axioms that capture the essential properties of the operations of conjugation in a group and algebraically encode the three Reidemeister moves from classical knot theory. This notion dates back to the early 1980's when Joyce and Matveev independently introduced the notion of a quandle and associated it to the complement of a knot. We will focus on an introduction to the theory of quandles by considering examples, discussing applications, and introducing recent work in this area.

Enhancements of Counting Invariants

1:50 PM - 2:30 PM
Sam Nelson, Claremont McKenna College

Quandle counting invariants form an infinite family of knot invariants which as easy to define and compute. Enhancements are stronger knot invariants defined using the quandle counting invariants as a base. In this talk we will see some examples of enhancements of quandle counting invariants.

An Introduction to Quandle Cohomology

2:40 PM - 3:20 PM
J. Scott Carter, University of South Alabama

The most easily defined invariants of a knot are related to the diagram of the knot to be colorable. Colorability conditions are given at the crossings of the diagram, and an associated algebraic structure called a quandle can be defined. The axioms of a quandle are derived from the Reidemeister moves. A similar structure and set of coloring conditions can be given for knotted trivalent graphs. The Reidemeister moves and their analogues suggest a secondary invariant that assigns certain values in an abelian group to crossings. The function values are cocycles in a specific homology theory.
In this talk, we will work from the ground up and demonstrate how the algebraic structures are necessitated by the geometric descriptions.

What is Categorification?

3:30 PM - 4:10 PM
Mikhail Khovanov, Columbia University

This talk will be an introduction to the idea of categorification. In categorification, numbers are lifted to vector spaces, while vector spaces equipped with integral lattices become Grothendieck groups of categories. Linear operators between vector spaces are lifted to functors. The talk will include several examples to motivate and illustrate categorification.

From Jones to Chebyshev: Adventures in Categorification

4:20 PM - 5:00 PM
Radmila Sazdanovic, North Carolina State University

Categorification is a method introduced at the end of the 20th century and successfully used in many branches of mathematics. Categorification realizes various mathematical objects as shadows of new, algebraically richer objects, a perspective that often leads to beautiful and structurally deep mathematics. A famous example is the Khovanov homology, which categorifies the Jones polynomial and has led to some of remarkable recent results in knot theory.
We will describe a related diagrammatic categorification of the ring of one-variable polynomials with integer coefficients. In particular, we will construct a diagrammatic algebra and use it to recover some well-known facts about the Chebyshev polynomials.